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If $$z=\frac{1}{2}-2 i$$ is such that $$|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}$$ and $$\alpha, \beta \in \mathbb{R}$$, then $$\alpha+\beta$$ is equal to

JEE · Math · previous-year question

  1. A.2
  2. B.$$-$$4
  3. C.3correct
  4. D.$$-$$1

Answer

C. 3

Explanation

To begin with, let's analyze the given equation: $$|z+1|=\alpha z+\beta(1+i)$$ First, we compute the modulus of the left side: Let's take the given value of z, $$z = \frac{1}{2} - 2i$$ Now, we find $$z + 1$$: $$z + 1 = \left(\frac{1}{2} - 2i\right) + 1 = \frac{3}{2} - 2i$$ Then, the modulus of $$z + 1$$ is calculated as follows: $$|z + 1| = \left|\frac{3}{2} - 2i\right| = \sqrt{\left(\frac{3}{2}\right)^2 + (-2)^2}$$ $$|z + 1| = \sqrt{\frac{9}{4} + 4}$$ $$|z + 1| = \sqrt{\frac{9}{4} + \frac{16}{4}}$$ $$|z + 1| = \sqrt{\frac{25}{4}}$$ $$|z + 1| = \frac{5}{2}$$ Now we need to equate the modulus to the right-hand side of the equation and solve for $$\alpha$$ and $$\beta$$. Let's rewrite the equation: $$\frac{5}{2} = \alpha z + \beta(1 + i)$$ Substitute $$z$$ with its value: $$\frac{5}{2} = \alpha \left(\frac{1}{2} - 2i\right) + \beta(1+i)$$ Rewrite the equation separating real and imaginary parts: $$\frac{5}{2} = \alpha \left(\frac{1}{2}\right) - 2\alpha i + \beta + \beta i$$ $$\frac{5}{2} = \left(\alpha \frac{1}{2} + \beta\right) + \left(-2\alpha + \beta\right)i$$ For the above equality to hold, both real and imaginary parts must be equal. Equating real parts: $$\alpha \frac{1}{2} + \beta = \frac{5}{2}$$ Equating imaginary parts: $$-2\alpha + \beta = 0$$ We now have a system of two linear equations: 1) $$\frac{\alpha}{2} + \beta = \frac{5}{2}$$ 2) $$-2\alpha + \beta = 0$$ Let's solve the system by isolating $$\beta$$ from the second equation and then substituting it into the first one: $$\beta = 2\alpha$$ Now substitute $$\beta$$ in the first equation: $$\frac{\alpha}{2} + 2\alpha = \frac{5}{2}$$ $$\alpha \left(\frac{1}{2} + 2\right) = \frac{5}{2}$$ $$\alpha \left(\frac{1}{2} + \frac{4}{2}\right) = \frac{5}{2}$$ $$\alpha \left(\frac{5}{2}\right) = \frac{5}{2}$$ To find the value of $$\alpha$$, we divide both sides by $$\frac{5}{2}$$: $$\alpha = 1$$ Now, we use the value of $$\alpha$$ to find $$\beta$$: $$\beta = 2\alpha$$ $$\beta = 2 \cdot 1$$ $$\beta = 2$$ Finally, we add both $$\alpha$$ and $$\beta$$ to find $$\alpha + \beta$$: $$\alpha + \beta = 1 + 2 = 3$$ The value of $$\alpha + \beta$$ is 3. So, the correct answer is Option C) 3.

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